Hatzalah help non-jews . If a non-jew calls and when they paramedics arrive and see a big cross, they will still help (Many non-jews calls them in NYC)
In Amish communities, they dont have health insurance, when an Amish gets sick the community raises the money to pay the medical bills
There is also christian Health Insurance, Meaning some christian organizations run a Health Insurance type plan. Meaning you pay money to help others and they will pay when you need it, However they wont pay for certain procedures they deem offensive (Like Abortions). They do accept jews if they want to participate as long as they follow their rules (Like no Abortions)
I suppose Jewish organizations could run something similar, but I dont think the community is big enough to support such a thing. Medical bills could run in to the millions and you are on the hook for them if someone gets really sick